You are given the following information obtained from a random sample of 4 observations.

25 47 32 56

At a .05 level of significance, determine whether or not the mean of the population from which this sample was taken is significantly different from 48. (Assume the population is normally distributed.)

To determine whether the mean of the population from which this sample was taken is significantly different from 48, we can conduct a t-test.

The steps for conducting a t-test are as follows:

Step 1: State the null hypothesis and the alternative hypothesis.
- Null hypothesis (H0): The population mean is equal to 48.
- Alternative hypothesis (H1): The population mean is not equal to 48.

Step 2: Calculate the sample mean and the sample standard deviation.
- Sample mean (x̄): calculated by adding up all the values and dividing by the sample size.
- Sample standard deviation (s): calculated by finding the square root of the variance of the sample.

For the given sample, the sample mean (x̄) is calculated by adding up all the values and dividing by the sample size:
x̄ = (25 + 47 + 32 + 56) / 4 = 40

The sample standard deviation (s) is calculated as follows:
s = √[(Σ(x - x̄)²) / (n - 1)]
= √[((25 - 40)² + (47 - 40)² + (32 - 40)² + (56 - 40)²) / (4 - 1)]
= √[(225 + 49 + 64 + 256) / 3]
= √(594 / 3)
≈ √198
≈ 14.07

Step 3: Calculate the test statistic.
- The test statistic for a t-test is calculated as (x̄ - μ) / (s / √n), where μ represents the population mean and n represents the sample size.

Given that μ = 48, n = 4, x̄ = 40, and s = 14.07, the test statistic is calculated as:
test statistic = (40 - 48) / (14.07 / √4)
= -8 / (14.07 / 2)
= -8 / 7.035
≈ -1.14

Step 4: Determine the critical value.
- The critical value is obtained from the t-distribution table or using statistical software, based on the desired level of significance (α) and the degrees of freedom (df).

For a two-tailed test at a 0.05 level of significance with 3 degrees of freedom (4-1), the critical value is approximately ± 3.182.

Step 5: Compare the test statistic with the critical value.
- If the absolute value of the test statistic is greater than the critical value, we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.

In this case, |test statistic| = |-1.14| ≈ 1.14, which is less than 3.182. Therefore, we fail to reject the null hypothesis.

Step 6: State the conclusion.
- Based on the sample data and the t-test, there is not enough evidence to conclude that the mean of the population from which this sample was taken is significantly different from 48 at a 0.05 level of significance.

To determine whether the mean of the population from which the sample was taken is significantly different from 48, we can conduct a hypothesis test. We will use the t-test for a single sample.

Here are the steps to perform the hypothesis test:

Step 1: State the null hypothesis and the alternative hypothesis:

Null hypothesis (H0): The mean of the population is equal to 48.
Alternative hypothesis (H1): The mean of the population is not equal to 48.

Step 2: Determine the level of significance (alpha):

The level of significance, also known as alpha (α), is the probability of rejecting the null hypothesis when it is true. In this case, the level of significance is given as .05.

Step 3: Calculate the test statistic:

We will calculate the t-test statistic using the formula:

t = (x̄ - μ) / (s / √n)

where x̄ is the sample mean, μ is the population mean (48 in this case), s is the sample standard deviation, and n is the sample size.

Given the sample: 25, 47, 32, 56
Sample mean (x̄) = (25 + 47 + 32 + 56) / 4 = 40
Sample standard deviation (s) = √[((25-40)^2 + (47-40)^2 + (32-40)^2 + (56-40)^2) / (4-1)] = 12.81
Sample size (n) = 4

Plugging these values into the t-test formula:
t = (40 - 48) / (12.81 / √4) = -1.56

Step 4: Determine the critical value:

The critical value is the value that separates the region of rejection from the region of acceptance. Since we are testing for a two-tailed hypothesis, we need to find the critical value for a t-distribution with a degree of freedom of n-1 (4-1=3) at the 0.025 significance level.

Using a t-distribution table or statistical software, we find that the critical value is approximately ±3.182.

Step 5: Make a decision:

If the test statistic falls within the region of rejection, we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.

Since |-1.56| < 3.182, the test statistic does not fall within the region of rejection.

Step 6: State the conclusion:

Based on the test results, we fail to reject the null hypothesis. There is not enough evidence to conclude that the mean of the population is significantly different from 48 at a 0.05 level of significance.

Z = (mean1 - mean2)/standard error (SE) of difference between means

SEdiff = √(SEmean1^2 + SEmean2^2)

SEm = SD/√(n-1)

If only one SD is provided, you can use just that to determine SEdiff.

Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion related to the Z score.